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Category: Problem-Solving

What Do You Notice? poster – Number Grid Puzzle

What Do You Notice? poster – Number Grid Puzzle

I hosted my first Family Math Night event of the school year last week. I had 33 student station facilitators – the most ever! It’s so exciting when kids volunteer to spend an evening doing math.

I also had a new What Do You Notice? poster. Here’s the skinny…

Title: Number Grid Pattern

Skills
K-2: number recognition, pattern
3-5: pattern, addition

Background Information: My youngest son visited the Basilica Sagrada Familia, a Roman Catholic Church in Barcelona, Spain and brought this pattern back for me as a gift. Here’s a photo of his gift: (And before you read the next paragraph where I describe the main pattern, you may want to discover your own patterns first.)

Having done a lot of these types of puzzles, it didn’t take me long to figure out the all rows, columns, and diagonals add to 33. It just so happens that 33 was the age of Jesus when he died.

The second and third columns are interesting. Notice how they both have the numbers ’14’ and ’10’. The second column includes ‘7’ and ‘2’. The third column includes ‘6’ which is one less than ‘7’ and ‘3’ which is one more than ‘2’. Number sense tells us that both columns, therefore, should add to the same number – which they do.

Younger students can focus on number recognition, repeated numbers, finding the number that represents their age, etc.

Here’s what it looked like at the event:

The Power of the Estimation Jar

The Power of the Estimation Jar

We often think of math as the exact-answer subject.  But the kind of math that we do most often during the day doesn’t require an exact answer.  We use this particular math skill when we need to figure out how much time we need to get ready in the morning.  Or whether we have enough gas in the car to get to work. Or whether $50 is enough to cover the items in our shopping cart.

The math skill we use the most is, of course, estimation. And estimating accurately requires a high level of math.  That’s because it’s abstract which means we need to tap into our number sense and reasoning skills.

One way to provide our students with opportunities to work on their estimation skills is during computation practice.  Instead of diving right in to figure out 15 x 12, have students come up with an estimate…about what the answer will be. In fact, periodically I ask students NOT to determine the exact answer and, instead, have them turn in their work with only their estimates recorded.  This is hard for them to do in the beginning because they are so used to working out arithmetic problems, but they soon learn the value in thinking about the problem first.

A fun way to get students to work on their estimating skills is through the estimation jar.  I’ve included two of my estimation videos below.  The first video describes using the estimation jar in the classroom as a way to develop, not only estimation skills, but place value and number sense, as well.

The second video is filled with tips on setting up your estimation table at your Family Math Night event.  It includes something I’ve been adding to my estimation tables recently – the use of a referent.

You’ll find in both videos that there is a heavy emphasis on getting students to think about and make sense of numbers.  I discovered an example of this in action one day while cleaning up after a Family Math Night event.  It was such a powerful example of number sense that I’m now including “thinking” paper at my estimation stations.  If you missed the newsletter where I describe this priceless find, check it out here.  And click here to get the pdf of the thinking paper I’m now using.

A Twist on the Estimation Jar – Classroom Version

Setting up the Estimation Table at your Family Math Night event

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Pattern and the Common Core State Standards in Mathematics

Pattern and the Common Core State Standards in Mathematics

Mathematics is often described as the science of pattern.  Through looking for, reasoning about, and describing numeric and geometric patterns, students come to realize that mathematics reflects order and predictability.  This is a significant discovery because students who understand the power of patterns in math are more confident in their ability to do math.  So when the Common Core State Standards first came out and I didn’t see a whole lot about pattern and patterning activities in the early years, I wondered why.

And I’ve been wondering why until recently when I read a fabulous article about teaching math.  The article was an interview with Bethany Rittle-Johnson, a professor of psychology and human development at Vanderbilt University in Tennessee.  Her studies focus on early math and the importance of teaching young children about patterns.  Here’s what she said about why pattern was not included in the standards:

“Patterns were mostly left out of the Common Core Math Standards in the early grades (kindergarten and 1st grade) due to a lack of evidence that they helped children understand later math concepts.”

But then she goes on to say that a lot of research since then proves that pattern should actually be included.  I agree.  In fact, I would argue that there had already been a lot of research underscoring the importance of teaching pattern in the early years yet for some reason, it was ignored.

Here’s why I think teaching pattern in those early years is important:

  • The study of pattern is the foundation of mathematics.  As I mentioned earlier, mathematics is described as the science of pattern.
  • It is the thread that binds all parts of mathematics together.
  • Discovering patterns makes life easier; patterns are predictable.*
  • Searching for patterns trains the brain to look critically.
  • Looking for patterns helps make connections between concepts in mathematics and other curricular areas.
  • Looking for patterns helps encourage students to be persistent and better problem solvers – they know there is predictability in mathematics and that mathematics makes sense.
  • Pattern can be used as a self-check device.
  • Patterns help students when they begin to make generalizations about number.

* to predict is to use known information to predict unknown information

Now, to be fair, the CCSSM Mathematical Practices (MP7 and MP8) do mention looking for patterns.  But pattern isn’t specifically called out in the content standards and I think that’s a mistake.  The word ‘pattern’ needs to be a part of the mathematical vocabulary so much so that looking for patterns becomes a natural part of what students do in math class.

Let me give you some examples.  All of the What Do You Notice? posters that I include in my Family Math Night events are perfect examples of looking for and describing patterns.  What I love about these posters is that they can be accessed on a variety of levels but all of those levels require looking for patterns.  In addition, some of the posters clearly show the connections between arithmetic and geometry making pattern the thread that binds all parts of mathematics together.

On a higher level, describing patterns helps lead us into making generalizations – the foundation for algebra.  By making generalizations, math changes from isolated bits and pieces to an organized and much more manageable body of information.  For example, through patterns, the numbers 1 through 100 are no longer 100 separate and isolated pieces of information to learn.  Instead, students simply need to learn 1 – 20 and then each of the decade names.

So we need to be doing pattern-specific activities in those early years.  Make AB patterns with teddy bears.  Sort blocks into different categories.  And always, always, always use the word ‘pattern’ when describing math.

By the way, I feel so strongly about pattern in the early grades that we devoted a station in our Nifty Numbers kit to it.  It’s important.  Without pattern, math simply does not exist.

Happy patterning~

 

The Estimation Jar – Number Sense in Action

The Estimation Jar – Number Sense in Action

I was cleaning up the Estimation Table at my last Family Math Night event when I noticed a slip of paper next to the Hershey’s jar.  Taking a closer look at it, I realized I was looking at the thinking behind someone’s guess as to the number of Hersheys in the jar.

estimation table slip

This piece of paper is priceless to me as an educator.  It allows me to clearly understand the steps this child took to arrive at his/her answer – an answer that turned out to be exactly two Hersheys kisses off!

It starts with a multiplication problem:  4 x 27.  It’s hard to see from this photo, but if you counted the number of Hersheys that can be seen on the side of the jar, I’m guessing this student got ’27’.  Then, if you look at the number of rows of 27 that could be made from one side of the jar to the other side, I’m guessing that that’s where the ‘4’ came from.  From there, the student knew s/he had to multiply the 27 Hersheys by the 4 rows.  Since, from this point forward the student uses addition, I’m going to guess that the student either wasn’t comfortable with double-digit by single-digit multiplication or simply did not know how to do it.

So, instead, s/he used number sense by breaking down 4 x 27 into a simpler problem:  (27 x 2) + (2 x 27) which s/he wrote as (27 + 27) + (27 + 27).  From there it was simply finding the answer of ’54’ and adding that twice to get 108.

This is an amazing example of a student that has a clear mastery of number sense – breaking a multiplication problem down into a more manageable addition problem.  It’s also a great example of the distributive property of multiplication, although there’s a good chance the student has no idea what ‘distributive property’ means.  It doesn’t really matter; it’s the concept that’s important.

And this is what I love about the estimation jar – it gives kids an opportunity to practice number sense within the context of something fun…candy.  And because there’s a sense of excitement and anticipation over who will get the closest and win all the candy, kids want to participate.

From now on I’m going to make sure I include scrap paper at all of my Estimation Tables.